Anyone seriously interested in Colin Maclaurin (1698-1746) or in eighteenth-century mathematical physics will welcome this book. For the first time, it brings into modern printed form three relatively rare “physical dissertations” by Maclaurin. These are: Maclaurin’s 1713 Glasgow thesis Concerning Gravity and Other Natural Forces; his 1724 Académie des Sciences de Paris prize-winning essay on the physics of impact, Demonstration of the Laws of Collisions of Bodies; and his 1740 Paris prize-winning essay on the tides Concerning the Physical Cause of the Flow and Ebb of the Sea, which includes significant contributions to the study of the shape of the earth. Tweddle has translated them accurately (the first from Latin, for which he includes the original text; the other two from French) into clear modern English. He tells us about Maclaurin’s sources and the linkage of his work with that of his contemporaries. And he explains passages and results that would be obscure to the modern reader, turning geometric or verbal material into analytical language and integrals when this helps.

What is there for nonspecialist readers? Maclaurin’s 1713 thesis lets us see how the youthful Maclaurin integrated his Newtonianism and his deep religious views, and how he advanced the Newtonian demolition of Descartes’ theory of vortices in the ether. Here is a sample quotation: “Gravity prevents mountains, seas, cities, people, and other living beings thrown off the surface of the earth from being scattered… the subsistence and nutrition of both humans and the other living beings depend on gravity; thus it is that the lord of the earth and the preserver of mankind is to be recognized most deservedly as the creator of gravity.” (p. 18)

The 1724 paper on impact gives a window into the early eighteenth-century dispute over what was the “force of a moving body” that is conserved when bodies collide. The Newtonians championed mv and Leibnizians championed “vis viva” or mv2. Maclaurin, though ingeniously arguing for mv, seems also to have voiced what became the standard later judgment, that this was a dispute about words. Tweddle convincingly argues that the most original part of the 1724 paper is the “nice geometrical treatment” of oblique collisions. Those who become interested in the vis viva controversy will find many suggestions for further reading in Tweddle’s notes and the comprehensive 118-item bibliography.

The 1740 essay on the tides is by far the most substantial of the three papers, but because so many of its achievements are included in Maclaurin’s 1742 Treatise of Fluxions and the later works it influenced, from Clairaut to Chandrasekhar, the major interest of this paper will be the amazing geometry used and Tweddle’s explanations of it. I especially liked Tweddle’s Appendix III, which gives geometric and analytic descriptions of the key properties of ellipses and then uses triple integrals to explicate Maclaurin’s calculations of the gravitation of spheroids (ellipses rotated on their axis). Leavening all this heavy mathematical physics is this quotation from Maclaurin’s 1740 paper: “Certainly it has to be admitted that the cause of gravity is not known, or at the very least it is obscure; however, bodies are not less heavy on that account.” (p. 100)

This book is not light reading. The commonplace that analytic geometry drove out the deep geometrical knowledge of the conic sections possessed by people like Newton and Maclaurin will be amply demonstrated to the reader of Maclaurin’s “fundamental theorem” (from the 1740 paper on the tides) on the equilibrium of a spheroidal mass of fluid whose parts attract one another according to Newton’s law of gravity (see, for instance, Maclaurin’s diagram on page 108). But the assiduous reader will be rewarded in many ways, both by working through Tweddle’s introductions, notes, and appendices, and by reading Maclaurin’s own words in Tweddle’s clear and accurate translations. I find the book refreshing in its old-fashioned scholarly virtues: instead of flashy and provocative superficialities, we have the result of years of profound study and deliberation, careful textual analysis, and sound understanding and explanation of the relevant mathematics and physics.

Judith V. Grabiner, the Flora Sanborn Pitzer Professor of Mathematics at Pitzer College, is the author of books and articles in the history of mathematics, including several recently on Colin Maclaurin.

General Introduction.- Part I: MacLaurin on Gravity.- Part II: MacLaurin on Collisions.- Part III: MacLaurin on the Tides.- References.- Index.